1. Discrete Mathematics for Computer Science
COMPSCI 230
Discrete Mathematics for Computer Science
Graphs

2. A tree is a connected graph with no cycles
What’s a tree?
A tree is a connected graph with no cycles

3. Tree

4. Not aTree

5. Not a Tree

6. Tree

7. How Many n-Node Trees? 1: 2: 3: 4: 5:

8. Notation In this lecture: n will denote the number of nodes in a graph
e will denote the number of edges in a graph

9. Theorem: Let G be a graph with n nodes and e edges
The following are equivalent:
1. G is a tree (connected, acyclic)
2. Every two nodes of G are joined by a unique path
3. G is connected and n = e + 1
4. G is acyclic and n = e + 1
5. G is acyclic and if any two non-adjacent points are joined by adding a new edge, the resulting graph has exactly one cycle

10. To prove this, it suffices to show
1  2  3  4  5  1

11. 1  2 1. G is a tree (connected, acyclic)
2. Every two nodes of G are joined by a unique path
Proof: (by contradiction)
Assume G is a tree that has two nodes connected by two different paths:
Then there exists a cycle!

12. 2  3 2. Every two nodes of G are joined by a unique path
3. G is connected and n = e + 1
Proof: (by induction)
Assume true for every graph with < n nodes
Let G have n nodes and let x and y be adjacent x y G1 G2
Let n1,e1 be number of nodes and edges in G1
Then n = n1 + n2 = e1 + e2 + 2 = e + 1

13. 3  4 3. G is connected and n = e + 1 4. G is acyclic and n = e + 1
Proof: (by contradiction)
Assume G is connected with n = e + 1, and G has a cycle containing k nodes
k nodes
Note that the cycle has k nodes and k edges
Starting from cycle, add other nodes and edges until you cover the whole graph
Number of edges in the graph will be at least n

14. Corollary: Every nontrivial tree has at least two endpoints (points of degree 1)
Proof:
Assume all but one of the points in the tree have degree at least 2
In any graph, sum of the degrees = 2e
Then the total number of edges in the tree is at least (2n-1)/2 = n - 1/2 > n - 1

15. How many labeled trees are there with three nodes?1 2 3 1 3 2 2 1 3

16. How many labeled trees are there with four nodes?c d

17. How many labeled trees are there with five nodes?5 4 3 5 4 2 5
labelings
labelings
labelings
125 labeled trees

18. How many labeled trees are there with n nodes?
3 labeled trees with 3 nodes
16 labeled trees with 4 nodes
125 labeled trees with 5 nodes
nn-2 labeled trees with n nodes

19. Cayley’s Formula
The number of labeled trees on n nodes is nn-2

20. The proof will use the correspondence principle
Each labeled tree on n nodes
corresponds to
A sequence in {1,2,…,n}n-2 (that is, n-2 numbers, each in the range [1..n])

21. How to make a sequence from a tree?
Loop through i from 1 to n-2
Let L be the degree-1 node with the lowest label
Define the ith element of the sequence as the label of the node adjacent to L
Delete the node L from the tree
Example: 8 5 3 4 1 7 2 6 1 3 3 4 4 4

22. How to reconstruct the unique tree from a sequence S:
Let I = {1, 2, 3, …, n}
Loop until S is empty
Let i = smallest # in I but not in S
Let s = first label in sequence S
Add edge {i, s} to the tree
Delete i from I
Delete s from S
Add edge {a,b}, where I = {a,b}

23. S: 1 3 3 4 4 4 I: 1 2 3 4 5 6 7 8 8 5 3 4 1 7 2 6

24. Spanning Trees
A spanning tree of a graph G is a tree that touches every node of G and uses only edges from G
Every connected graph has a spanning tree

25. A graph is planar if it can be drawn in the plane without crossing edges

26. Examples of Planar Graphs=

27. Faces
4 faces
A planar graph splits the plane into disjoint faces

28. Euler’s Formula
If G is a connected planar graph with n vertices, e edges and f faces, then n – e + f = 2

29. Rather than using induction, we’ll use the important notion of the dual graph
Dual = put a node in every face, and an edge between every adjacent face

30. n = eT + 1 n + f = eT + eT* + 2 f = eT* + 1 = e + 2
Let G* be the dual graph of G
Let T be a spanning tree of G
Let T* be the graph where there is an edge in dual graph for each edge in G – T
Then T* is a spanning tree for G*
n = eT + 1
n + f = eT + eT* + 2
f = eT* + 1
= e + 2

31. Corollary: Let G be a simple planar graph with n > 2 vertices. Then:
1. G has a vertex of degree at most 5
2. G has at most 3n – 6 edges
Proof of 1 (by contradiction):
In any graph, (sum of degrees) = 2e
Assume all vertices have degree ≥ 6
Then e ≥ 3n
Furthermore, since G is simple, 3f ≤ 2e
So 3n + 3f ≤ 3e, 3n + 3f = 3e + 6, and 3e + 6 ≤ 3e

32. Graph Coloring
A coloring of a graph is an assignment of a color to each vertex such that no neighboring vertices have the same color

33. Graph Coloring Arises surprisingly often in CS
Register allocation: assign temporary variables to registers for scheduling instructions. Variables that interfere, or are simultaneously active, cannot be assigned to the same register

34. Theorem: Every planar graph can be 6-colored
Proof Sketch (by induction):
Assume every planar graph with less than n vertices can be 6-colored
Assume G has n vertices
Since G is planar, it has some node v with degree at most 5
Remove v and color by Induction Hypothesis

35. Not too difficult to give an inductive proof of 5-colorability, using same fact that some vertex has degree ≤ 5
4-color theorem remains challenging!

36. Implementing Graphs

37. Adjacency Matrix
Suppose we have a graph G with n vertices. The adjacency matrix is the n x n matrix A=[aij] with:
aij = 1 if (i,j) is an edge
aij = 0 if (i,j) is not an edge
Good for dense graphs!

38. Example
A =

39. Counting Paths
The number of paths of length k from node i to node j is the entry in position (i,j) in the matrix Ak
A2 = =

40. Adjacency List
Suppose we have a graph G with n vertices. The adjacency list is the list that contains all the nodes that each node is adjacent to
Good for sparse graphs!

41. Example 1 2 3 4
1: 2,3
2: 1,3,4
3: 1,2,4
4: 2,3

42. Here’s What You Need to Know…
Trees
Counting Trees
Different Characterizations
Planar Graphs
Definition
Euler’s Theorem
Coloring Planar Graphs
Adjacency Matrix and List
Useful for counting
Here’s What You Need to Know…